| import Mathlib |
| import OAI.Analysis.HardSphere.PathBounds.LatticeExponentialSum |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology |
| open MeasureTheory Set |
| open scoped ENNReal |
| namespace HardSphere |
| open MeasureTheory Set Filter |
|
|
| lemma finset_separated_net {ι : Type*} [DecidableEq ι] (A : Finset ι) (d : ι → ι → ℝ) |
| (hd : ∀ i j, d i j=d j i) (hdiag : ∀ i, d i i=0) {R : ℝ} (hR : 0<R) : |
| ∃ B : Finset ι, B⊆A ∧ (∀ i∈B, ∀ j∈B, i≠j → R≤d i j) ∧ |
| ∀ i∈A, ∃ j∈B, d i j<R := by |
| classical |
| let P : Finset ι → Prop := fun B ↦ ∀ i∈B, ∀ j∈B, i≠j → R≤d i j |
| let S := A.powerset.filter P |
| have hS : S.Nonempty := ⟨∅,by simp [S,P]⟩ |
| obtain ⟨B,hBS,hmax⟩ := S.exists_max_image Finset.card hS |
| have hBA : B⊆A := Finset.mem_powerset.mp (Finset.mem_filter.mp hBS).1 |
| have hBP : P B := (Finset.mem_filter.mp hBS).2 |
| refine ⟨B,hBA,hBP,?_⟩ |
| intro i hi |
| by_contra hn |
| have hall : ∀ j∈B, R≤d i j := by |
| intro j hj |
| by_contra hlt |
| exact hn ⟨j,hj,lt_of_not_ge hlt⟩ |
| have hin : i∉B := by |
| intro hiB |
| have hh := hall i hiB |
| rw [hdiag] at hh |
| exact (not_le_of_gt hR) hh |
| have hPi : P (insert i B) := by |
| intro j hj k hk hjk |
| rcases Finset.mem_insert.mp hj with hji | hjB |
| · subst j |
| rcases Finset.mem_insert.mp hk with hki | hkB |
| · exact (hjk hki.symm).elim |
| · exact hall k hkB |
| · rcases Finset.mem_insert.mp hk with hki | hkB |
| · subst k |
| rw [hd j i] |
| exact hall j hjB |
| · exact hBP j hjB k hkB hjk |
| have hmem : insert i B∈S := Finset.mem_filter.mpr ⟨Finset.mem_powerset.mpr (Finset.insert_subset hi hBA),hPi⟩ |
| have hh := hmax (insert i B) hmem |
| rw [Finset.card_insert_of_notMem hin] at hh |
| omega |
|
|
| theorem unitGrid_separated_subset : ∃ Q : ℕ, 0<Q ∧ |
| ∀ (K : ℕ) (L : Length), L.val=(K:ℝ) → ∀ A : Finset (CubeCell K), |
| ∃ B : Finset (CubeCell K), B⊆A ∧ |
| (∀ v∈B, ∀ w∈B, v≠w → 20≤torusDistance L (unitCenter L v) (unitCenter L w)) ∧ |
| (A.card:ℝ)≤(Q:ℝ)*(B.card:ℝ) := by |
| obtain ⟨C,hC,hbound⟩ := unitGrid_exponential_bound |
| let Q : ℕ := ⌈Real.exp 20*C⌉₊+1 |
| have hQ : 0<Q := Nat.succ_pos _ |
| have hQB : Real.exp 20*C≤(Q:ℝ) := by |
| exact (Nat.le_ceil _).trans (by dsimp [Q]; push_cast; linarith) |
| refine ⟨Q,hQ,?_⟩ |
| intro K L hL A |
| obtain ⟨B,hBA,hB,hcover⟩ := finset_separated_net A |
| (fun v w ↦ torusDistance L (unitCenter L v) (unitCenter L w)) |
| (fun v w ↦ torusDistance_comm L _ _) (fun v ↦ by simp [torusDistance]) (by norm_num : (0:ℝ)<20) |
| refine ⟨B,hBA,hB,?_⟩ |
| have hpoint (v : CubeCell K) (hv : v∈A) : |
| 1≤Real.exp 20*∑ w∈B, Real.exp (-torusDistance L (unitCenter L v) (unitCenter L w)) := by |
| obtain ⟨w,hw,hd⟩ := hcover v hv |
| calc |
| _ ≤ Real.exp (20-torusDistance L (unitCenter L v) (unitCenter L w)) := |
| Real.one_le_exp (by linarith only [hd]) |
| _ = Real.exp 20*Real.exp (-torusDistance L (unitCenter L v) (unitCenter L w)) := by |
| rw [sub_eq_add_neg,Real.exp_add] |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (Finset.single_le_sum (f:=fun w : CubeCell K ↦ Real.exp (-torusDistance L (unitCenter L v) (unitCenter L w))) |
| (fun _ _ ↦ (Real.exp_pos _).le) hw) (Real.exp_pos 20).le |
| calc |
| _ = ∑ _v∈A, (1:ℝ) := by simp |
| _ ≤ ∑ v∈A, Real.exp 20*∑ w∈B, Real.exp (-torusDistance L (unitCenter L v) (unitCenter L w)) := |
| Finset.sum_le_sum hpoint |
| _ = Real.exp 20*∑ w∈B, ∑ v∈A, Real.exp (-torusDistance L (unitCenter L v) (unitCenter L w)) := by |
| rw [←Finset.mul_sum,Finset.sum_comm] |
| _ ≤ Real.exp 20*∑ _w∈B, C := by |
| apply mul_le_mul_of_nonneg_left _ (Real.exp_pos 20).le |
| apply Finset.sum_le_sum |
| intro w _ |
| exact (Finset.sum_le_univ_sum_of_nonneg (fun v : CubeCell K ↦ (Real.exp_pos (-torusDistance L (unitCenter L v) (unitCenter L w))).le)).trans (hbound K L hL _) |
| _ = (Real.exp 20*C)*(B.card:ℝ) := by simp; ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right hQB (Nat.cast_nonneg _) |
|
|
| end HardSphere |
|
|
| end |
|
|
| end OAI |
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|
